Three Constructions
Introduction
This activity shows the three geometric constructions used by Ptolemy and others in ancient Greece to model the motion of planets along the ecliptic. Ptolemy included all three in most of his models of the planets in order to account for their retrograde motions and their apparent changes in speed along the ecliptic.
In each animation below, we see the planet in the ecliptic plane. Its daily motion with the celestial sphere is neglected.
Epicycle
Description
The center of the Epicycle orbits the Earth on the Deferent circle at a uniform speed defined by the Speed slider. The Planet itself travels about the Epicycle. The two motions together account for the apparent motion of the planet along the ecliptic which is shown by the point Planet's Zodiac. This point is defined, as in the above constructions, by a line from the Earth through the planet but is not shown here for the sake of clarity.
Ptolemy used this construction to account for the retrograde motion of planets. By adjusting the relative size of the epicycle compared to the deferent and the speed of the planet on the epicycle, Copernicus used this construction as a replacement for the equant which he and his contemporaries disliked.
Usage
The Epicycle Size slider defines the radius of the epicycle compared to the deferent circle.
The Epicycle Period slider defines the time it takes the planet to circle the epicycle relative to the time it takes the epicycle to circle the deferent.
When the value for Epicycle Period is less than the value for Epicycle Size, the planet will undergo retrograde motion. In other cases, this construction will only effect the apparent speed of the planet as it travels along the ecliptic.
The Animate checkbox will start and stop the animation. The Show Trace checkbox will show the actual path of the planet within the model. Changing the value of the Epicycle Size, Epicycle Period, or Size sliders will clear these tracings.
The Size slider defines the radius of the deferent. As the epicycle radius is defined relative to the deferent radius, adjusting this slider will scale the size of the entire planet's model. Doing so has no effect on the apparent motion of the planet along the ecliptic. This demonstrates that a geocentric model of the planets had no way of determining the size of different planets' orbits.
Eccentric
Description
The planet is driven around the deferent circle at a uniform speed. However, the center of the deferent circle, labeled 'O', is displaced from the Earth. A line from the Earth through the Planet shows the resulting apparent motion of the Planet along the Ecliptic.
The effect on the apparent motion from this construction is qualitatively similar to that of the equant, but different in its detail. This construction was also used by Copernicus in his heliocentric models of the planet.
If we consider the Planet shown as the Sun, this construction can be used to account for the unevenness of the seasons.
Usage
The Point labeled O can moved by drag-and-drop. Locations further from the Earth will have a more dramatic effect. Positioning O at different angular coordinates will effect which parts of its tropical cycle the Planet appears to move faster or slower.
The length of the Planet's tropical period in days can be entered in the Tropical Period field.
The Tropical Cycle Time shows how many days since the Planet was directly right of O until its current position along the deferent.
The Ecliptic Position shows the angular position of the Planet along the Ecliptic.
The position of the Planet along the Deferent can be manually changed using the Deferent Angle Slider.
The Animate checkbox will start and stop the animation. The Show Trace checkbox will periodically mark the apparent position of the planet along the ecliptic (Planet's Zodiac). When these marks are further apart, this indicates a faster apparent motion of the planet. Changing the position of the point O will clear the tracing.
Equant
Description
The planet is driven around its deferent circle at a uniform speed as seen by the Center of Motion (CoM). The actual motion of the planet is then faster when it is further from the Center of Motion. A line from the Earth through the Planet shows the resulting apparent motion of the planet along the ecliptic.
This construction was later criticised by Copernicus and his contemporaries as an unwelcome compromise to Aristotelian Physics and was regarded as unphysical.
Usage
The Point labeled CoM can moved by drag-and-drop. Locations further from the Earth will have a more dramatic effect. Positioning CoM at different angular coordinates will effect which parts of its tropical cycle the Planet moves faster or slower.
The length of the Planet's tropical period in days can be entered in the Tropical Period field.
The Tropical Cycle Time shows how many days since the Planet was directly to the right of CoM until its current position along the deferent.
The Ecliptic Position shows the angular position of the Planet along the Ecliptic.
The position of the Planet along the Deferent can be manually changed using the Deferent Angle Slider.
The Animate checkbox will start and stop the animation. The Show Trace checkbox will periodically mark the apparent position of the planet along the ecliptic (Planet's Zodiac). When these marks are further apart, this indicates a faster apparent motion of the planet. Changing the position of the CoM will clear the tracing.
All constructions
Description
This final model includes all three constructions and shows how the planet's apparent motion would be observed with all three constructions included.
The primary use of this is to demonstrate the complicated nature of Ptolemy's Task. Each slider had to be appropriately adjusted so that the motion of the planet along the ecliptic would match observations of actual planets.
In The Almagest, the models for each planet were based on this model, but sometimes additional complications were needed such as placing an additional epicycle on the epicycle. The parameters for each planet's model were chosen based on observation alone without any uniform guiding principle as to why the parameters should take certain values.
Usage
This model works similarly to the others above with some exceptions. As was the case in Ptolemy's models, the Equant is positions so that it is equally distant from the Eccentric as the Earth but on the opposite side. The Eccentric slider is redefined compared to the other versions to insure that the Earth and the Center of Motion are never found within the epicycle circle. Setting the value to zero will minimize its effect on the apparent motion. Similarly, setting the value to one will maximize the effect.